Nora is creating a dog run in her backyard for her dog, Max. She has enough materials to create a run with a perimeter of 120 feet. This is represented by the function A=−w2+60w

Question

𝐴
=
−
𝑤
2
+
60
𝑤
, where A
𝐴
is the area of the run and w is the width. Which of the following best analyzes the meaning of the x-intercept?(1 point)
Responses

The dog run must have a width between 0 and 900 feet.
The dog run must have a width between 0 and 900 feet.

The dog run must have a width between 0 and 30 feet.
The dog run must have a width between 0 and 30 feet.

The dog run must have a width between 0 and 60 feet.
The dog run must have a width between 0 and 60 feet.

The dog run must have a width between 0 and 120 feet.
The dog run must have a width between 0 and 120 feet.

Answer 1

To analyze the x-intercept of the function \( A = -w^2 + 60w \), we need to understand what the x-intercept represents in the context of the problem.

The x-intercept occurs where the area \( A \) is equal to zero. Setting the area function equal to zero gives us:

\[ -w^2 + 60w = 0 \]

Factoring this equation, we get:

\[ w(w - 60) = 0 \]

From this, we find the solutions \( w = 0 \) and \( w = 60 \).

The x-intercept of 0 means that if the width \( w \) of the dog run is 0, then it does not exist (i.e., the area is zero), and the x-intercept of 60 means that if the width is 60 feet, the area will also be zero (i.e., the length will be constrained such that there is no area when the width is at the maximum permissible width, since the perimeter is fixed).

Thus, the x-intercepts indicate that the width of the dog run must lie between 0 and 60 feet for the area to be positive.

Considering the options provided, the correct interpretation of the x-intercept is:

The dog run must have a width between 0 and 60 feet.